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On an Online Spanning Tree Problem in Randomly Weighted Graphs

Published online by Cambridge University Press:  14 August 2006

JAN REMY
Affiliation:
Institut für Theoretische Informatik, ETH Zürich, Switzerland (e-mail: jremy@inf.ethz.ch, asouza@inf.ethz.ch, steger@inf.ethz.ch)
ALEXANDER SOUZA
Affiliation:
Institut für Theoretische Informatik, ETH Zürich, Switzerland (e-mail: jremy@inf.ethz.ch, asouza@inf.ethz.ch, steger@inf.ethz.ch)
ANGELIKA STEGER
Affiliation:
Institut für Theoretische Informatik, ETH Zürich, Switzerland (e-mail: jremy@inf.ethz.ch, asouza@inf.ethz.ch, steger@inf.ethz.ch)

Abstract

This paper is devoted to an online variant of the minimum spanning tree problem in randomly weighted graphs. We assume that the input graph is complete and the edge weights are uniformly distributed over [0,1]. An algorithm receives the edges one by one and has to decide immediately whether to include the current edge into the spanning tree or to reject it. The corresponding edge sequence is determined by some adversary. We propose an algorithm which achieves $\mathbb{E}[ALG]/\mathbb{E}[OPT]=O(1)$ and $\mathbb{E}[ALG/OPT]=O(1)$ against a fair adaptive adversary, i.e., an adversary which determines the edge order online and is fair in a sense that he does not know more about the edge weights than the algorithm. Furthermore, we prove that no online algorithm performs better than $\mathbb{E}[ALG]/\mathbb{E}[OPT]=\Omega(\log n)$ if the adversary knows the edge weights in advance. This lower bound is tight, since there is an algorithm which yields $\mathbb{E}[ALG]/\mathbb{E}[OPT]=O(\log n)$ against the strongest-imaginable adversary.

Type
Paper
Copyright
© 2006 Cambridge University Press

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